Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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1answer
21 views

central series of $\frac{G}{Z_2(G)}$

Let $G$ be finite p-group I am trying to make central series for $\frac{G}{Z_2(G)}$ and more inportant what is nilpotency class of $\frac{G}{Z_2(G)}$ since ...
2
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1answer
61 views

Set of all inner automorphisms is a normal subgroup

In order to prove this, I first proved that the set of all automorphisms from a group $G$ to $G$ form a group under composition: The identity homorphism is an automorphism because sends $x$ from $G$ ...
-4
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0answers
31 views

assume $ M/N $ be a chief factor of $ G $. Why $ M/N $ has prime order or order $ 4 $?

Let $ G $ is a soluble group and $ \Phi(G) $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Let $ K/L $ be a chief factor of $ G $. Let $ M $ be the smallest normal ...
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0answers
21 views

minimal normal subgroup of a chief factor of soluble group $ G $ is a minimal normal normal subgroup of $ G $?

Let $ G $ is a soluble group and $ \Phi(G) $. Let $ K/L $ be a chief factor of $ G $. Let $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent and $ M \neq 1 $. assume $ M/N $ ...
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0answers
20 views

Let $ K/L $ be a chief factor of $ G $ and $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent

Let $ G $ is soluble group with $ \Phi(G) = 1 $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Let $ K/L $ be a chief factor of $ G $ and $ M $ be the smallest normal ...
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0answers
17 views

IS $ F(G) $ a direct product of some minimal normal subgroups of G?

Let $ G $ is a finite group and $ F(G) $ is the fitting subgroup of $ G $. IS $ F(G) $ a direct product of some minimal normal subgroups of G? Why ? $ F(G) $ is the largest nilpotent normal subgroup ...
1
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1answer
31 views

Let $ H/N $ is nilpotent quotient subgroup. Then is $ N $ characteristic in $ H $? If no, then what condition need?

Let $ G $ is finite solvable group and $ H $ is normal subgroup. Let $ H/N $ is nilpotent quotient subgroup. Then is $ N $ characteristic in $ H $? If no, then what condition need?
3
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1answer
52 views

Action of ${\rm Aut}(G)$ on $G$

Let $G$ be a finite group and consider the natural action of ${\rm Aut}(G)$ on $G$ and let there are two orbits under this action. How could we show that $G$ is an (elementary) abelian group? Is ...
2
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1answer
25 views

Problems with P. Hall theorem proof (The problem involves the use of Frattini's argument)

Theorem Let G be a solvable group of order $ab$, where $(a,b)=1$. Then $G$ contains at least one subgroup of order $a$, and any two such are conjugate. Details The proof the book presents involves ...
2
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1answer
21 views

How many subgroups are there in an elementary-$p$ group

$C_p\times C_p$ has $1+1+(p-1)=p+1$ subgroups of order $p$ (all of which $\cong C_p$), so it has $1+(1+p)+1=3+p$ subgroups totally. But how many subgroups in $C_p\times C_p \times C_p\times C_p$? ...
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0answers
57 views

Is the monster group a characteristic quotient of $F_2$?

Let $F_2$ be the free group on two generators, and $M$ the monster group. It's known that every finite simple group is 2-generated, so let $F_2\rightarrow M$ be a surjection with kernel $N$. Let ...
0
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1answer
30 views

A group of order 2p (p prime) and other conditions - prove abelian.

I have G where $|G|=2p$ ; p is prime. $\exists a\in Z\left(G\right);\:a^2=e$. I need to prove that G is abelian. Now, let's translate it into math. To prove that G is abelian, is in other words ti ...
1
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1answer
40 views

p-element centralizing a Sylow p-subgroup

Let $G$ be a finite group, $P$ a Sylow $p$-subgroup for a prime $p$ and $g$ a $p$-element with $gxg^{-1} = x$ for all $x \in P$. Then $g \in Z(P)$. Is this true? How can i prove that $g \in P$ ...
1
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1answer
32 views

Without using Cauchy's or Sylow theorems ; can we prove that every group of order $65$ is cyclic? [duplicate]

Without using Cauchy's or Sylow theorems, can we prove that every group of order $65$ is cyclic? Please help, thanks in advance (any technique of group homomorphisms and normal subgroups can be used). ...
2
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0answers
49 views

I have a finite permutation group and access to a computer algebra system, how can I recognize the structure of the group?

I have a fixed permutation group $G$, and cannot tell which finite group it really is in a "human readable" way. I also have GAP. Is there a step by step computation to give me the structure of this ...
0
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1answer
42 views

Let $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent. What's mean smallest normal subgroup?

theorem: Let $ G $ be solvable with $ \Phi(G)=1 $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Then every chief factor of $ G $ has prime order or is $ G $-isomorphic ...
0
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1answer
53 views

Dilemma with the classification theorem of finite groups

We know that if $H < G$ , $G$ commutative, and $G/H \cong \hat H < G$, then $ G \cong H \oplus \hat H$. Then on the basis of this I could write $Z_4 \cong Z_2 \oplus Z_2 $ but we know that $Z_4 ...
2
votes
0answers
30 views

p-group of class 2 with cyclic commutator subgroup

I was reading a paper "ON THE EXISTENCE OF NONINNER AUTOMORPHISMS OF ORDER TWO IN FINITE 2-GROUPS" you can get it from here. In Introduction page 2 He said "It is worth mentioning here that we need ...
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1answer
26 views

G's order is a multiply of coprime numbers, need to prove about its subgroups.

dont want you to answer me directly, only a direction of thinking. I have abelian $G$ of finite order $np : p>n, p>1,$ and p is prime. $A,B\le G$ are sub-groups of G of order $p$ both. I need ...
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0answers
27 views

$ G = MC $ for some cyclic subgroup $ C $. If $ \vert G : M \vert = 4 $ then why $ G \cong S_{4} $?

Let $ M $ is a maximal subgroup of finite group $ G $, that $ G = MC $ for some cyclic subgroup $ C $. If $ \vert G : M \vert = 4 $ and $ M_{G} = 1 $ then why $ G\cong S_{4} $?
2
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0answers
21 views

$G$ be a finite group and $G'$ be its commutator subgroup and $\widehat G$ be the character group of $G$ ; then $G/G' \cong \widehat G$?

Let $G$ be a finite group and $G'$ be its commutator subgroup and $\widehat G$ be the character group of $G$ ; then is it true that $G/G' \cong \widehat G$ ?
2
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1answer
59 views

Pretty easy equations of elements in a group

Problem $G$ is a group generated by $a,b\in G$ such that $a^5=e$, $aba^{-1}=b^2$ and $b\ne e$. I want to find the order of $b$. Attempt I tried to multiply the second equation from right by ...
0
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1answer
56 views

Every minimal normal subgroup is contained in the center

G is a finite group in the following questions: (X):Every minimal normal subgroup is contained in the center. (1) Let $N$ and $M$ be normal subgroups of $G$, both of which satisfy (X), then prove: ...
0
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1answer
23 views

Another question about a proof that a group is abelian

G is a group, $A,B\le G$ : $$x^{-1}Ax\in A$$ $$x^{-1}Bx\in B$$ And I already proved that AB is a sub-group and $x^{-1}ABx\in AB$ Now I need to prove that $$ab = ba$$ for any $a\in A, b\in B$ and I ...
1
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2answers
58 views

Ordered pairs of permutations in symmetric group

How many ordered pairs $\left(\alpha_1,\alpha_2\right)$ of permutations in symmetric group $S_n$ that commute: $$\alpha _1 \circ \alpha _2 = \alpha _2 \circ \alpha _1\,,$$ where $\alpha _1, \alpha _2 ...
1
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1answer
20 views

Order of multypling 2 sub-groups who's orders are coprime

Im given as an exercize to prove that an order of 2 sub-groups A,B who's orders are coprime, is: $$|A| \cdot |B|$$ What I know that generally: $$|AB|=\frac{\left|A\right|\cdot ...
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0answers
16 views

characterization of all finite subgroups of mobius transformation

does there exist any characterization of all finite subgroups of the group $G$, where $G$ is the group of automorphisms of the open unit disk ?
1
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1answer
12 views

conditions on multypling sub-groups (so it would be a new sub-group)

Ok, so we know that if G is abelian and A,B are here sub-groups, then AB, defined by: $AB\:=\:\left\{ab\::\:a\in A,\:b\in B\right\}$ is a new sub-group. Now, I'm given another condition and I need ...
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0answers
32 views

Prove that if $G/Z(G)$ is powerful group then $G$ is of class $2$

Is following statement is true? If yes, then how we can prove it? If $G$ is finite nonabelian $p$-group and $\frac{G}{Z(G)}$ is powerful group then $G$ is of nilpotency class $2$.
4
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2answers
79 views

$G$ finite, the number of distinct conjugates of $x$ is the index of the normalizer $N_x$ of $\{x\}$ in $G$

In order to prove this, I did the following: first, I showed that conjugacy forms an equivalence relation, then I can find its conjugacy classes. I understand how to form a conjugacy class, given a ...
2
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1answer
20 views

$ M_{G} $ is the core of $ M $ in $ G $. Let $ M_{G} = 1 $. Why $ M $ is a complement for $ N $ in $ G $?

Let $ M $ maximal subgroup of solvable group $ G $, and assume that $ G = MC $ for some cyclic subgroup $ C $. Let $ N $ be a minimal normal subgroup of $ G $, then $ N $ is an elementary abelian $ p ...
-1
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1answer
17 views

$ N $ is an elementary abelian $ p $-subgroup. Is true $ N $ is cyclic group? $

Let $ G $ is solvable group and $ M $ be a maximal subgroup of $ G $. Let $ N $ be a minimal normal subgroup of $ G $, then $ N $ is an elementary abelian $ p $-subgroup. Is true $ N $ is cyclic ...
3
votes
3answers
130 views

$G$ has order $p^a$, then the center of $G$ counts more than the identity

This question makes no sense to me. I don't know what it means by "counts more than the identity". Then, the exercise gives me a hint: "break G into equivalence classes of conjugacy elements and see ...
4
votes
1answer
47 views

Prove $[N(H):H]\equiv [G:H](\mod p)$

Let $G$ be a finite group and $H$ be a subgroup of $G$. Let $|H|=p^n$ for some $p$ prime, $n\geq1$. Show that $[N(H):H]\equiv [G:H](\mod p)$. I observed that I will need to show that $p$ divides ...
2
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1answer
34 views

Picking out a subset of elements from a finite product of cyclic groups

Let $C_n$ be the cyclic group of order $n$, and let $G = \prod_{i=1}^n C_n = \underbrace{C_n \times C_n \times \ldots \times C_n}_{n \text{ times}}$. For $g = (g_1,g_2,\ldots, g_n) \in G$, call $g$ ...
0
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0answers
41 views

Is it true $ N(P \cap M) = NP \cap NM = N(NC) \cap G = P \cap G = P $?

Let $ M $ be a maximal subgroup of a solvable group $ G $, and assume that $ G = MC $, for some cyclic Subgroup $ C $. Let $ N $ be a minimal normal subgroup of $ G $. Then $ N $ is an elementary ...
1
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1answer
58 views

Assume that $ G = MC $, for some cyclic subgroup $ C $. Is $ M \cap C $ a normal subgroup of $ G $?

Let $ G $ is a solvable finite group and $ M $ be a maximal subgroup of $ G $, and assume that $ G = MC $, for some cyclic subgroup $ C $. If $ M_{G} = 1 $ that $ M_{G} $ is core of $ M $ in $ G $, ...
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0answers
34 views

Let $ M $ be a maximal subgroup of a solvable group $ G $, and assume that $ G=MC $, for some cyclic subgroup $ C $.

Let $ M $ be a maximal subgroup of a solvable group $ G $, and assume that $ G=MC $, for some cyclic subgroup $ C $. Then $ \vert G : M \vert $ is a prime or $ 4 $. Also if $ \vert G : M \vert = 4 $ , ...
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3answers
86 views

Show that $G$ is cyclic

Here is a problem from Herstein. Let G be a finite abelian group so that the equation $x^n=e$ has at most $n$ solutions in $G$ for every positive integer $n$. Show that $G$ is cyclic. I will ...
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2answers
61 views

The difference between Z(G) and C(a) in an example

I found that I didnt understood the defenitions. I have this exercize: to prove that $a\in Z(G)$ $<==>$ $C(a) = G$ Is there here something to prove? Isnt it directly of their defenitions ? I ...
2
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1answer
46 views

Is this subset of GL(C) a sub-group?

$$G\:=\:GL_2\left(R\right)$$ $$\:N=\left\{A\in G\:;\:A\cdot A^T=I\right\}$$ Is N a subgroup of G? So the main work here is to prove closure for the opposite. We want $A^{-1}\in ...
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1answer
54 views

Does there exist a group of order $ p^\alpha q^\beta$ that is simple?

I am working on an problem that calls for the existance of simple group of order $p^\alpha q^\beta$ with $\alpha , \beta \geq 1$. I was wondering if such a group existed. Edit: The problem I was ...
1
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1answer
25 views

G finite group with the following characteristics

G is closed finite group with an associative operator with the following parametres/characterstics: For each $x,y\in G$ if $ax = ay$ then $x = y$ for each $a\in G$ For each $w,z\in G$ if $wa = za$ ...
0
votes
3answers
60 views

G is finite group. Need to proof that exists natural k that $g^k = e$ [duplicate]

How do I prove that in a finite group G, for each element in G there is natural power (say $k$) which depends on g,such that $g^k=e$ ? I need to show the existence and the dependence on which $g$ I ...
1
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1answer
39 views

Unit cancellation in group rings

Suppose I have a finite group $G$, a non-trivial proper subgroup $H$, a field $k$ (restricting to $k=\mathbb C$ would be fine), and non-zero elements $a,u$ in the group algebra $kG$ satisfying the ...
4
votes
1answer
45 views

Normal group that contains its centralizer

I am studying for my Algebra qual and I came across this question: Let $G$ be a finite group with a normal subgroup $N$ such that $C_G (N) \leq N$. Show that $$ |G|\leq |N|!. $$ Here $C_G (N)$ is ...
1
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2answers
32 views

Let $G$ be a group of all $2 \times 2$ matrices over $Z_p$ with determinant $1$ under matrix multiplication. To find the order of $G$.

I am solving some previous year's question paper of our college and found the following problem: Let $p$ be a prime number. Let $G$ be a group of all $2 \times 2$ matrices over $Z_p$ with determinant ...
1
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2answers
39 views

A generator of intersection of two sub-groups.

How do I find it? For example, an easy one: $G\:=\:\left(\mathbb{Z},+\right)$ H and K are sub-groups: $n\mathbb{Z}, m\mathbb{Z}$ for different $n$ and $m$. And we know that $n$ and $m$ are ...
1
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1answer
15 views

Can we embed every finite group in some special orthogonal group or special linear group of some order , over $\mathbb R$?

For every finite group $G$ , does there exist $n \in \mathbb Z^+$ such that $G$ can be embedded in $SO_n(\mathbb R)$ ? Can every finite group be embedded in $SL_n(\mathbb R)$ for some $n$ ?
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1answer
25 views

Finite Cyclic Group - for each x : x in pow of its order is identity unite

We know that, if G is finite cyclic group, then for each $x\in G$, $$x^{\left|G\right|}=e$$ So I have an easy exercize, to show that $U_8$ isnt cyclic (Without lagrange or something more ...